Nuprl Lemma : Rice-theorem-for-Type_2
∀F:Type ⟶ 𝔹. ((∀X,Y:Type. (X ~ Y ⇒ F X = F Y)) ⇒ (∀X,Y:Type. (F X = F Y ∨ (∀f:ℕ ⟶ 𝔹. Dec(∀n:ℕ. f n = ff)))))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
nat: ℕ,
bfalse: ff,
bool: 𝔹,
decidable: Dec(P),
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
exists: ∃x:A. B[x],
and: P ∧ Q,
nat-inf: ℕ∞,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
squash: ↓T,
nat2inf: n∞,
less_than: a < b,
sq_type: SQType(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
bnot: ¬bb,
nat-inf-infinity: ∞
Lemmas referenced :
Rice-theorem-for-Type_1,
all_wf,
nat_wf,
bool_wf,
decidable_wf,
equal-wf-T-base,
exists_wf,
nat-inf_wf,
not_wf,
assert_wf,
nat2inf_wf,
nat-inf-infinity_wf,
equal_wf,
equipollent_wf,
bnot_wf,
b-exists_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
assert_of_bnot,
assert-b-exists,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
int_seg_properties,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
itermSubtract_wf,
intformeq_wf,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
set_wf,
less_than_wf,
primrec-wf2,
decidable__exists_int_seg,
decidable__equal_bool,
btrue_wf,
subtract-add-cancel,
iff_imp_equal_bool,
lt_int_wf,
assert_of_lt_int,
iff_wf,
subtype_base_sq,
bool_subtype_base,
true_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
assert_functionality_wrt_uiff,
bfalse_wf,
assert_elim,
btrue_neq_bfalse
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
unionElimination,
inlFormation,
isectElimination,
functionEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
baseClosed,
inrFormation,
productElimination,
productEquality,
universeEquality,
cumulativity,
instantiate,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
addLevel,
impliesFunctionality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
imageMemberEquality,
imageElimination,
impliesLevelFunctionality,
equalityElimination,
hyp_replacement
Latex:
\mforall{}F:Type {}\mrightarrow{} \mBbbB{}
((\mforall{}X,Y:Type. (X \msim{} Y {}\mRightarrow{} F X = F Y))
{}\mRightarrow{} (\mforall{}X,Y:Type. (F X = F Y \mvee{} (\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. Dec(\mforall{}n:\mBbbN{}. f n = ff)))))
Date html generated:
2017_10_01-AM-08_29_44
Last ObjectModification:
2017_07_26-PM-04_24_07
Theory : basic
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